Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-3a^2 - 48a - 189}{a^3 + 6a^2 - 7a}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-3(a^2 + 16a + 63)} {a(a^2 + 6a - 7)} $ $ z = -\dfrac{3}{a} \cdot \dfrac{a^2 + 16a + 63}{a^2 + 6a - 7} $ Next factor the numerator and denominator. $ z = - \dfrac{3}{a} \cdot \dfrac{(a + 7)(a + 9)}{(a + 7)(a - 1)}$ Assuming $a \neq -7$ , we can cancel the $a + 7$ $ z = - \dfrac{3}{a} \cdot \dfrac{a + 9}{a - 1}$ Therefore: $ z = \dfrac{ -3(a + 9)}{ a(a - 1)}$, $a \neq -7$